Ramsey theorems for knots, links and spatial graphs

Abstract

An embedding f: G —► R of a graph G into R is said to be linear if each edge f(e) (e G E(G)) is a straight line segment. It will be shown that for any knot or link type k , there is a finite number R(k) such that every linear embedding of the complete graph Kn with at least R(k) vertices (n > R(k)) in R contains a knot or link equivalent to k .